if the average atomic mass of an element having two isotopes is 71,the ratio of the natural abundance of the two isotopes of mass 70 and 74 will be

To find the ratio of the natural abundances of two isotopes with masses of 70 and 74, when the average atomic mass is given as 71, we can set up a simple equation based on the definition of average atomic mass. Let's break it down step by step.

Conceptual Understanding of Isotopes

Isotopes are variants of a particular chemical element that have the same number of protons but different numbers of neutrons. This means they share the same atomic number but have different atomic masses. In this case, we have two isotopes of an element: one with a mass of 70 and another with a mass of 74.

Setting Up the Equation

Let’s define the natural abundances of the isotopes:

• Let x be the fraction of the isotope with mass 70.
• Then, (1 - x) will represent the fraction of the isotope with mass 74.

The average atomic mass (A) can be calculated using the formula:

A = (mass of isotope 1 * abundance of isotope 1) + (mass of isotope 2 * abundance of isotope 2)

Plugging in the values we have:

71 = (70 * x) + (74 * (1 - x))

Solving the Equation

Now, let’s solve for x:

71 = 70x + 74 - 74x

Combining like terms gives us:

71 = 74 - 4x

Now, isolate x:

4x = 74 - 71

4x = 3

x = 3/4

This means the fraction of the isotope with mass 70 is 3/4, and the fraction of the isotope with mass 74 is:

1 - x = 1 - 3/4 = 1/4.

Finding the Ratio

Now we can express the abundances as a ratio:

The ratio of the natural abundances of the two isotopes is:

Ratio (70:74) = (3/4):(1/4)

To simplify, we can multiply both sides by 4, resulting in:

Ratio = 3:1.

Conclusion

This means for every three atoms of the isotope with mass 70, there is one atom of the isotope with mass 74. So, the ratio of the natural abundance of the two isotopes is 3:1.

This example illustrates how average atomic mass reflects the weighted contributions of isotopes based on their natural abundances, helping us understand the composition of elements in nature.